Contributions à l'Etude du Vertex Topologique en Théorie ... - Toubkal

326 L.B. Drissi et al. / Nuclear Physics B 804 [PM] (2008) 307–3415. Field model for O(−3) → E (t,∞)We first study the gauge invariant supersymmetric field model with target space given by thecurve E (t,∞) = ∂P 2 . For details on the degenerate elliptic curve E (t,∞) ,seeAppendix A. Then,we consider the extension to O(−3) → E (t,∞) .5.1. Divisors of local P 2To begin, notice that the local P 2 Eq. (4.7) has several divisors; i.e., codimension one subspacesdescribing boundary patches of the normal bundle of the projective plane.The standard ones are obtained by setting one of the z i ’s to zero; z i = 0 with i = 0, 1, 2, 3.5.1.1. Toric boundary of P 2In this paragraph, we consider the three following complex surfaces [D 1 ], [D 2 ] and [D 3 ],[D 1 ]: |z 2 | 2 +|z 3 | 2 − 3|z 0 | 2 = t ⇔ z 1 = 0,[D 2 ]: |z 3 | 2 +|z 1 | 2 − 3|z 0 | 2 = t ⇔ z 2 = 0,[D 3 ]: |z 1 | 2 +|z 2 | 2 − 3|z 0 | 2 = t ⇔ z 3 = 0,and their union [D]=[D 1 ]∪[D 2 ]∪[D 3 ].To see what this local geometry describes precisely; let us set |z 0 | 2 = 0 in above equationsfrom where one sees that each relation describes a complex one dimension projective line P 1 .Todistinguish between these complex projective lines, we use the convention notation P 1 iwhere thesubindex i refers to z i = 0. Thus we have(5.1)P 1 1 : |z 2| 2 +|z 3 | 2 = t,P 1 2 : |z 3| 2 +|z 1 | 2 = t,P 1 3 : |z 1| 2 +|z 2 | 2 = t.(5.2)As we see, these projective lines have the following intersection matrix 6( )−2 1 1P 1 i ∩ P1 j = 1 −2 1 ,(5.3)1 1 −2from which one sees that the complex curveE (t,∞) = P 1 1 ∪ P1 2 ∪ P1 3 ,is elliptic (E (t,∞) ∼ T 2 ). Indeed, computing3∑E (t,∞) ∩ E (t,∞) = P 1 i ∩ P1 i + 2( P 1 1 ∩ P1 2 + P1 2 ∩ P1 3 + P1 3 ∩ 1) P1 ,i=1we get, up on using (5.3),E (t,∞) ∩ E (t,∞) =−3 × 2 + 2 × 3 = 0.(5.4)(5.5)(5.6)6 Denoting by {α 1 ,α 2 ,α 3 },abasisofH 2 (P 2 ,R), and by {A 1 ,A 2 ,A 3 }, the dual basis of H 2 (P 2 ,R)with ∫ A i α j = δ i j ,the intersection matrix Eq. (5.3) is given by I ij = ∫ P 2 α i ∧ α j .